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https://hdl.handle.net/2142/86980
Description
Title
A Variation on a Theme of Vasconcelos
Author(s)
Smith, Daniel Aaron
Issue Date
1999
Doctoral Committee Chair(s)
Griffith, Phillip A.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
In 1967, Vasconcelos and Ferrand independently demonstrated that an ideal I in a Noetherian local ring R is generated by a regular sequence if and only if I/ I2 is free over R/I and pdRI < infinity. In this thesis, we prove that a radical ideal I in an excellent local normal domain is generated by a regular sequence provided R/I satisfies the Serre condition S2, I/ I(2) is R/I-free, and I is generated by a regular sequence when localized at primes P containing I such that hat P/I ≤ 1. In particular, we are able to replace the strong assumption pdRI < infinity with essentially weaker hypotheses. Our proof makes use of lifting as presented by Auslander, Ding and Slobbered [1993] together with a finer analysis of the elimination of the obstructions to lifting cyclic modules. Subsequently, we develop some criteria for lifting a module M in a more general setting. Our approach here is to again assume the conclusion holds when localizing at primes of low codimension while also requiring M and a related module of morphemes to possess sufficient depth in the remaining cases.
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